What is the integral of 3/(5+2x^2) ?

The integral of 3/(5+2x^2) is 3/(sqrt(10))arctan(sqrt(2/5)x)+C

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integral of 3/(5+2x^2)

\int \frac{3}{5 + 2 x ^{2}} d x

\frac{3}{10} arctan (\frac{2}{5} x) + C

Solution steps

\int \frac{3}{5 + 2 x ^{2}} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \int \frac{1}{5 + 2 x ^{2}} d x

Apply Integral Substitution

= 3 \cdot \int \frac{1}{10 ( 1 + u ^{2} )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 3 \cdot \frac{1}{10} \cdot \int \frac{1}{1 + u ^{2}} d u

Use the common integral:

\int \frac{1}{1 + u ^{2}} d u = arctan (u)

= 3 \cdot \frac{1}{10} arctan (u)

Substitute back

u = \frac{2}{5} x

= 3 \cdot \frac{1}{10} arctan (\frac{2}{5} x)

Simplify

3 \cdot \frac{1}{10} arctan (\frac{2}{5} x) : \frac{3}{10} arctan (\frac{2}{5} x)

= \frac{3}{10} arctan (\frac{2}{5} x)

Add a constant to the solution

= \frac{3}{10} arctan (\frac{2}{5} x) + C

What is the integral of 3/(5+2x^2) ?
The integral of 3/(5+2x^2) is 3/(sqrt(10))arctan(sqrt(2/5)x)+C